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Plucking models of business
cycle uctuations: evidence
from the G-7 countries
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Department of Economics
BUSINESS CYCLE VOLATILITY AND ECONOMIC GROWTH
RESEARCH PAPER NO. 00-3
PLUCKING MODELS OF BUSINESS CYCLE
FLUCTUATIONS: EVIDENCE FROM THE G-7
COUNTRIES
TERENCE C. MILLS AND PING WANG
August 2000
This paper forms part of the ESRC funded project (Award No. L1382511013)
“Business Cycle Volatility and Economic Growth: A Comparative Time Series
Study”, which itself is part of the Understanding the Evolving Macroeconomy
Research programme.
2
Abstract
Friedman’s ‘plucking’ model, in which output cannot exceed a ceiling
level but is occasionally plucked downward by recessions, is tested using
Kim and Nelson’s formal econometric specification on output data from
the G-7 countries. Considerable support for the model is obtained,
leading us to conclude that during normal periods, output seems to be
driven mostly by permanent shocks, but during recessions and highgrowth recoveries, transitory shocks dominate. During these periods
macroeconomic models that emphasise demand-oriented shocks, rather
than real business cycle type models, may thus be more appropriate.
Keywords: BUSINESS CYCLE ASYMMETRY, STATE SPACE MODELS, G-7
COUNTRIES, MARKOV SWITCHING.
1.
Introduction
The recent interest in business cycle asymmetry has brought back into fashion
the “plucking model” suggested by Milton Friedman more than 30 years ago.
Friedman (1964, 1993) noticed a striking asymmetry in the correlations between
succeeding phases of the business cycle – the amplitude of contractions is strongly
correlated with succeeding expansions, but the amplitude of these expansions is
uncorrelated with the amplitude of succeeding contractions. He thus proposed a
model of business cycles in which he likened the path of output to a string attached to
the underside of a board, which may be held at an angle to represent secular growth.
The board represents an upper limit, or ceiling, to output, while the string is plucked
downward by recessionary shocks at irregular intervals. The extent of the output
decline will vary across recessions, but output will always rebound to the ceiling
level, so that recessions can only have temporary effects on output. Friedman’s
proposal has resulted in numerous studies of asymmetry, but a formal test of the
plucking model has only recently been provided by Kim and Nelson (1999a), in
which a concise survey of asymmetric business cycle research may also be found.
Existing attempts to incorporate asymmetry into time series models of
economic fluctuations typically have been limited to the growth rate of output or to its
trend component. Observing this, Kim and Nelson (1999a) develop a model that is
able to estimate the importance of downward shocks to both trend and cycle, and to
test the plucking hypothesis against a symmetric trend-plus-cycle alternative such as
that proposed by Clark (1987). The feature of their model is that it incorporates
asymmetric movements of business fluctuations away from trend and asymmetric
persistence of shocks during recessionary and normal times. They show that the
stochastic behaviour of quarterly U.S. output is well characterised by Friedman’s
plucking model, i.e., output is occasionally plucked down by recession and the
cyclical or transitory component exhibits asymmetric behaviour.
Most research on business cycle asymmetry is based on data from the U.S.,
although Goodwin and Sweeney (1993) focus on eight OECD countries. There have
been several recent studies of the U.K. business cycle, notably Artis and Zhang
(1999), Birchenhall et al (1999), Simpson et al (2000) and Öcal and Osborn (2000).
These have applied Markov-switching or smooth transition models to examine growth
in the two states of recession and expansion, but they do not decompose output into a
trend and a cycle in the two states, or investigate business cycle asymmetry in the
sense of plucking. To further our understanding of asymmetric output movements,
we test Friedman’s plucking model by extending Kim and Nelson’s (1999a) work to
the G-7 countries and present cross-country comparisons.
4
2.
Model Specification
The novel feature of Kim and Nelson’s specification is that they allow first-order
Markov switching variables in both the trend and cycle components, thus allowing
both plucking and asymmetric fluctuations around a stochastic trend. Consider the
following unobserved component model in which the logarithm of output, y t , is
decomposed into trend, τ t , and transitory, ct , components:
yt = τ t + ct
(1)
To allow for regime shifts or asymmetric deviations of y t from its trend, it is assumed
that innovations to the transitory component are a mixture of two different types of
shocks:
ct = φ1ct −1 + φ 2 ct − 2 + π St + ut ,
π St = πS t , π ≠ 0,
ut ~ N (0, σ u2,st ) ,
σ u2,st = σ u2, 0 (1 − S t ) + σ u2,1 S t ,
S t = 0 or 1 .
(2)
Here π S t is an asymmetric discrete shock, which is dependent upon an unobserved
variable, S t , while u t is the usual symmetric shock. S t is an indicator variable that
evolves according to a first-order Markov-switching process as in Hamilton (1989):
P [S t = 1 | S t −1 = 1] = p
P [S t = 0 | S t −1 = 0] = q
During ‘normal times’, S t = 0 and the economy is near to potential or trend output.
During ‘recessions’, however, S t = 1 and the economy is hit by a transitory shock,
possibly with a negative expected value if π < 0 . If this is the case, then aggregate
demand or other disturbances are ‘plucking’ output down. Equation (2) allows for the
possibility that the variance of the shock may be different in the two regimes. In the
absence of such shocks, ct evolves as a second order autoregressive process and
hence can display both persistence and pseudo-cyclical behaviour.
The specification of the trend component in (1) is consistent with Friedman’s
(1993) suggestion that potential output may be approximated by a random walk, with
various types of disturbances producing perturbations to it:
τ t = g t −1 + τ t −1 + vt ,
g t = g t −1 + wt ,
5
(
)
vt ~ N 0 , σ v2,St ,
(
)
wt ~ N 0 , σ w2 ,
σ v2,S t = σ v2,0 ( 1 − S t ) + σ v2,1 S t ,
The stochastic trend τ t is thus subjected to two shocks - to its level, through vt , and
to its growth rate, through wt . Specifying trend growth, g t , to be stochastic allows
the possibility of, for example, productivity slowdowns in output. The above
specifications again allow for the possibility that the variance of the level shock may
be different across regimes. On the other hand, the variance of g t is assumed not to
be systematically different across regimes.
In the transitory component, the term π St is incorporated to account for
potential asymmetric behaviour. In addition, if σ u2, St = 0 with π < 0 , this implies the
existence of a trend ceiling or maximum feasible output, as suggested in Friedman’s
plucking model. Thus, the regime characterised by S t = 1 and π < 0 would be
interpreted as a period during which the economy is being plucked down below its
trend ceiling level.
The model can be given the state-space representation
τ t 
c 
yt = [1 1 0 0]  t  = Hξ t
ct −1 
 
gt 
 0  1 0 0
π  0 φ φ
1
2
ξ t =  St  + 
 0  0 1 0
  
 0  0 0 0
Observation equation
1
vt 
u  _

0
ξ t −1 +  t  = µ st + Fξ t −1 + Vt
0 
0
 

1
 wt 
State equation
where
E (VtVt ' ) = QSt
and
σ v2,St

0
QS t = 
 0

 0
0
σ
2
u ,S t
0
0
0
0 

0
0 
0
0 

0 σ w2 ,St 
Estimation can be carried out by approximate maximum likelihood (Kim, 1994) using
the GAUSS routines supplied by Kim (see Kim and Nelson, 1999a, 1999b, chapter 5).
6
3.
Data and Results
We analyse the logarithms of quarterly real GDP (output) for the G-7
countries. All of the data come from Datastream, which in turn takes the data from
different sources. The sample periods are as follows: Canada, 1957:1 – 1999:4;
France, 1965:1 – 1999:4; Germany, 1960:1 – 1999:4; Italy, 1970:1 – 1999:4; Japan,
1955:2 – 1999:4; U.K., 1955:1 - 1999:4; U.S., 1950:1 – 2000:1.
Table 1 summarises the results for the U.S. and the U.K., where model 1 is the
unrestricted version described above and models 2 and 3 are restricted versions that
impose σ w = 0 and σ u2,1 = σ u2, 0 = 0 , respectively.
Looking first at the results for the U.S., we find that, not surprisingly, they are
very similar to those reported by Kim and Nelson (1999a), although they use a
somewhat different sample period to that employed here (they use 1951:1 – 1995:3).
The sum of the autoregressive cycle coefficients, φ1 + φ 2 , for the transitory
component is 0.7147, a little smaller than that reported by Kim and Nelson, (0.7969),
suggesting that once output is plucked down by negative transitory shocks, their
effects decay relatively faster after incorporating recent data. The estimated
coefficients yield a pair of complex autoregressive roots implying a period of 11
quarters. Comparing the likelihood values for the two models, the LR test statistics
for the hypothesis σ w = 0 is 3.77, rejecting the null at a p-value of 0.052, a slightly
stronger rejection than Kim and Nelson, for whom the test statistic was 3.44. We thus
conclude that U.S. trend growth has not been constant. Like Kim and Nelson, the LR
statistic for the joint hypothesis that σ u2,1 = σ u2, 0 = 0 allows us to accept the null at a
very high p-value (0.30), thus implying that the discrete shock explains most of the
dynamics in the transitory component. During normal times, the economy is thus
subject to mostly permanent shocks and is operating near its trend ceiling. During
recessions and the recovery periods that follow, however, the transitory component
plays a major role in producing output fluctuations. Note that πˆ = −0.01 and is
clearly significantly negative.
The estimated model is thus consistent with Kim and Nelson (1999a) and their
interpretation bears repeating. The estimate q̂ = 0.9554 implies that the expected
duration of a ‘normal’ regime is 22 quarters, while p̂ = 0.9214 implies an expected
duration of recession of 13 quarters. The steady state probabilities of being in a
normal or recessionary regime are 0.64 and 0.36, respectively. (The expected
duration of a normal regime, for example, is given by 1 (1 − q ) , while the steady state
probability is (1 − p ) (2 − p − q ) ). The onset of a recession thus produces a sequence
of negative transitory shocks which ‘pluck’ output down, although with φ1 + φ 2 =0.71,
their effects decay reasonably quickly. Near the end of the recession, and in the
absence of further negative shocks, the fast-decaying negative shocks give rise to a
7
third, high-recovery phase, moving the economy back near the trend ceiling again.
There is thus, à la Sichel (1994), three distinctive phases of business cycle dynamics:
a normal, a recessionary, and a high growth, recovery phase.
Figure 1(a), which shows y t and filtered estimates of the trend ceiling
component, τ t , reveals that most of the time the U.S. economy is operating at or near
to the trend ceiling. Kim and Nelson, 1999b, provide details of the construction of
filtered estimates: the first five years of each sample are omitted from these plots so
that we can focus on filtered estimates that have ‘settled down’. Figures 1(b) and 1(c)
show filtered estimates of the transitory component, ct , and the probability of a
negative shock to the transitory component, i.e., the probability that S t = 1
conditional on information up to t − 1 .
Table 1 Estimated models for U.S. and U.K.
U.S.
U.K.
Parameter
Model 1
Model 2
Model 3
Model 1
Model 2
Model 3
p
0.9330
0.9271
0.9214
0.9706
0.9708
0.9708
(0.0452)
(0.0428)
(0.0442)
(0.0194)
(0.0192)
((0.0190)
0.9649
0.9393
0.9554
0.9531
0.9535
0.9535
(0.0254)
(0.0367)
(0.0252)
(0.0368)
(0.0354)
(0.0352)
1.3285
1.5643
1.3189
1.4723
1.4770
1.4770
(0.1266)
(0.1084)
(0.1095)
(0.2314)
(0.2074)
(0.2021)
-0.6138
-0.7125
-0.6350
-0.6860
-0.6916
-0.6916
(0.1181)
((0.0861)
(0.1298)
(0.2246)
(0.1991)
(0.1942)
0.0014
0.0001
-
0.0003
0.0000
-
(0.0013)
(0.0034)
(0.0021)
(0.0000)
0.00005
0.00004
0.0000
0.0000
(0.0013)
(0.0012)
(0.0003)
(0.0001)
0.0050
0.0051
0.0051
0.0031
0.0032
0.0032
(0.0009)
(0.0005)
(0.0007)
(0.0004)
(0.0003)
(0.0003)
0.0115
0.0113
0.0115
0.0120
0.0120
0.0120
(0.0013)
(0.0010)
(0.0013)
(0.0008)
(0.0008)
(0.0008)
0.0008
-
0.0008
0.0000
-
0.0000
(0.0004)
(0.0000)
q
φ1
φ2
σu,0
σu,1
σv,0
σv,1
σw
(0.0004)
π
Log L’lihood
-
-
(0.0000)
-0.0099
-0.0062
-0.0098
-0.0039
-0.0039
-0.0039
(0.0027)
(0.0015)
(0.0027)
(0.0022)
(0.0021)
(0.0020)
607.359
605.473
607.211
528.842
528.840
528.840
8
The transitory component and the probability of negative shocks correlate
highly with NBER recessionary periods, as discussed in Kim and Nelson (1999a). The
figures also show that, apart from a brief period in the early 1990s (no doubt a
consequence of the Gulf war and the accompanying oil price shocks), the economy
has been running at ceiling output since 1985. This is consistent with the view that
this is the longest period of sustained expansion in U.S. economic history, typically
attributed to the fact that macroeconomic policy has kept inflation firmly under
control (Romer, 1999).
For the U.K. the sum of the AR coefficients is φ1 + φ 2 = 0.7863 , a little higher
than for the U.S. The estimated coefficients again imply complex roots with a period
of 13 quarters. Testing σ w = 0 by comparing models 1 and 2 produces an LR
statistic of 0.004, which therefore strongly supports the hypothesis that the variance of
trend growth rate is zero. This suggests that the trend component is only subject to
level shocks and that over this period trend growth has been constant, unlike the U.S.
Comparing models 1 and 3 provides a test of the joint hypothesis that σ u2,1 = σ u2, 0 = 0 .
The LR statistic of 0.004 obviously accepts the hypothesis, implying that, as in the
U.S., discrete shocks explain most of the dynamics in the transitory component. The
estimate of the impact of such shocks is πˆ = −0.004 , which is again negative but
rather smaller than for the U.S., while the estimates of p̂ and q̂ imply expected
durations of 34 and 22 quarters for recession and normal phases respectively, with
steady state probabilities of 0.61 and 0.39. The implications of these estimates are
seen in Figure 2. Figures 2(a) and 2(b) show that, unlike the U.S., output has
consistently been below its ceiling level for the great majority of the sample period.
Only since the mid-1990s has a lengthy period of ‘normal’ growth been achieved,
which corresponds well with contemporary views of the behaviour of the U.K.
economy (Bean and Crafts, 1996). The last five years of the sample are clearly
unusual, for Figure 2(c) shows that only the mid-1960s approximate in terms of the
prolonged absence of negative shocks.
Table 2 reports unrestricted estimates of the model for the other G-7 countries,
while Table 3 reports estimates of restricted models, where insignificant parameters
have been set to zero. All restrictions are accepted at very high p-values, as can be
seen by comparing the respective log-likelihoods in the two tables. Figures 3 to 7
present similar sets of plots to those presented for the U.S. and the U.K.
Compared with the U.S. and the U.K., these models provide a variety of
interpretations. Like the U.S., but unlike the U.K., estimates of q are always greater
than p, implying that recessions have shorter expected durations and lower steady
state probabilities than normal phases. These probabilities vary from 0.37 for Japan
to 0.83 for Germany, and all are much lower than for the U.S. and the U.K. Estimates
9
of shock persistence, φ1 + φ 2 , range from 0.5886 for Germany to 0.9458 for Canada,
and for Japan and France the transitory components exhibit exponential rather than
pseudo-cyclical behaviour. Only for France and Italy can the ‘trend ceiling’
hypothesis σ u2,1 = σ u2, 0 = 0 be accepted. For Japan and Canada only σ u2,1 = 0 , so that
just ‘normal’ symmetric shocks effect the transitory component, whereas the situation
is reversed for Germany, which is affected only by symmetric recessionary shocks.
On the other hand, regime dependent level shocks are observed for all countries, and
only for Germany is σ w = 0 , so that, like the U.K., trend growth has remained
constant. The evidence in favour of ‘plucking’ is somewhat mixed – although π is
always negative, it is not significantly so for Japan or Germany.
Table 2 Estimated unrestricted models for the other G-7 countries
Parameters
Canada
France
Italy
Germany
Japan
p
0.6144
0.7312
0.5539
0.8257
0.2945
(0.3359)
(0.1970)
(0.1604)
(0.1186)
(0.2447)
0.9866
0.9439
0.9786
0.9716
0.8974
(0.0138)
(0.0758)
(0.0242)
(0.0249)
(0.1194)
1.5411
0.4964
1.4931
-0.0234
1.0827
(0.2520)
(0.4467)
(0.1404)
(0.1062)
(0.1842)
-0.5852
0.3363
-0.6116
0.5223
-0.2196
(0.2253)
(0.3776)
(0.1675)
(0.1748)
(0.1870)
0.0043
0.0015
0.0013
0.00001
0.0069
(0.0021)
(0.0034)
(0.0023)
(0.0002)
(0.0012)
0.00001
0.0057
0.00001
0.0083
0.00002
(0.0003)
(0.0049)
(0.0004)
(0.0041)
(0.0002)
0.0073
0.0056
0.0051
0.0093
0.0001
(0.0013)
(0.0016)
(0.0009)
(0.0008)
(0.0018)
0.0047
0.0054
0.0034
0.0161
0.0180
(0.0024)
(0.0066)
(0.0008)
(0.0049)
(0.0059)
0.0006
0.0012
0.0008
0.0003
0.0015
(0.0004)
(0.0005)
(0.0003)
(0.0004)
(0.0005)
-0.0145
-0.0081
-0.0103
-0.0128
-0.0030
(0.0044)
(0.0036)
(0.0029)
(0.0087)
(0.0051)
490.94
418.23
365.59
427.32
511.20
q
φ1
φ2
σu,0
σu,1
σv,0
σv,1
σw
π
Log L’lihood
10
Table 3 Estimated restricted models for the other G-7 countries
Parameters
Canada
France
Italy
Germany
Japan
p
0.6143
0.6871
0.5582
0.8339
0.3728
(0.2229)
(0.2781)
(0.1509)
(0.1186)
(0.2870)
0.9866
0.9305
0.9824
0.9774
0.9275
(0.0138)
(0.1195)
(0.0201)
(0.0232)
(0.0870)
1.5411
0.8557
1.5458
_
0.9004
(0.2399)
(0.1035)
(0.1271)
-0.5953
_
-0.6705
0.5886
(0.1404)
(0.1677)
_
_
q
φ1
φ2
(0.2149)
σu,0
0.0043
_
(0.0654)
(0.0020)
σu,1
_
_
0.0072
(0.0012)
_
_
0.0095
_
(0.0050)
σv,0
σv,1
σw
π
Log L’lihood
0.0073
0.0059
0.0054
0.0097
_
(0.0013)
(0.0010)
(0.0005)
(0.0008)
0.0047
0.0091
0.0035
0.0170
0.0184
(0.0023)
(0.0033)
(0.0007)
(0.0046)
(0.0067)
0.0006
0.0010
0.0008
_
0.0017
(0.0003)
(0.0005)
(0.0003)
-0.0145
-0.0068
-0.0094
(0.0042)
(0.0038)
(0.0021)
490.94
417.96
365.58
(0.0006)
_
_
426.11
509.84
Figure 3 shows that Canadian output has been close to its ceiling value except
for two major episodes – throughout most of the 1980s and, like the U.S. and the
U.K., during the early 1990s. The recesssions are seen to be a consequence of two
major negative shocks to the transitory component of output. Canada was particularly
affected in the early 1980s by shifts in international trade patterns and also by labour
market rigidities caused by provincial disparities and structural differences. As noted
above, Canada has the greatest shock persistence of all the G-7 countries, and this is
seen in the length of the two recessions.
From Figure 4, we see that France has been consistently below its output
ceiling, with the large deviations in the 1970s being a consequence of it adapting only
with difficulty to wage and oil shocks. The 1981-83 decline was related to the
11
succession of exchange rate crises that eventually led to a major shift in
macroeconomic policy by the Mitterand administration, who then followed more
market-oriented policies which produced a more stable growth path (Sicsic and
Wyplosz, 1996).
By comparison, Germany has tended to be rather closer to trend, although with
some rather volatile fluctuations during the late 1960s and early 1970s (see Figure 5).
These were probably a consequence of the recession of 1966-67, which produced a
fall in employment as well as in output. This, in turn, led to significant changes in
work practices and a subsequent loss of discipline by both unions and employers
associations, manifesting itself in wild-cat strikes, which were exacerbated by an
appreciation of the exchange rate. Such instability ended after the floating of the
Deutschmark in 1973 and the adoption by the Bundesbank of a tight nonaccommodating monetary policy (see Carlin, 1996).
Italy, whose sample period only begins in 1970, was well below its ceiling
until the mid-1980s, but since then has been very close to trend except for 1992 to
1995 (see Figure 6). The poor performance during late 1970s and early 1980s
undoubtedly had its origins earlier, for the social and institutional developments of the
previous decade, which led to a succession of weak government coalitions, made the
Italian economy more vulnerable to exogenous shocks, particularly as the country was
also hit by waves of terrorism. The stable performance throughout the 1980s was a
reflection of a new commitment to exchange rate stability, industrial restructuring and
improved labour relations. Nevertheless, government borrowing remained out of
control and this precipitated the poor performance of the early 1990s (Rossi and
Toniolo, 1996).
Finally, from Figure 7, Japan has always been close to trend, but this masks
the fact that throughout the first half of the 1990s the transitory component has been
negative, reflecting the long recession after the stock market crash at the start of the
decade. As output moved back to its ceiling level in 1997, a large negative shock
(associated with the major tax increase in that year) tipped the economy back into
recession again, in which it has remained.
4.
Conclusions
Friedman’s plucking model, as rigorously formulated by Kim and Nelson (1999a,b),
offers an interesting approach to investigating asymmetries in business cycles. On
fitting this model to output data from the G-7 countries, we have found a fair degree
of support for it. Negative asymmetric discrete shocks are found to influence the
transitory component of output of each country, so that there does seem to be
evidence for the downward recessionary ‘plucks’ envisaged by Friedman (1964,
12
1993). The evidence of a ceiling level for output is less widespread, being accepted
for the U.S., the U.K., France and Italy, but not for Canada, Japan and Germany. The
hypothesis of constant trend growth can only be accepted for the U.K. and Germany
and typically both normal and recessionary symmetric shocks influence trend output.
The deviations of actual from estimated trend output for each country accords
well with macroeconomic history, although the length and depth of recessions vary
widely and only for the U.K. is the expected duration of a recession longer than that
of a ‘normal’ period. Nevertheless, our results for the wider set of G-7 economies are
consistent with those found by Kim and Nelson (1999a) for the U.S. Output during
normal times seems to be driven mostly by permanent shocks, but during recessions
and high-growth recoveries, transitory shocks dominate, so that during these periods
macroeconomic models that emphasise demand-oriented shocks, rather than real
business cycle type models, may be more appropriate.
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15
Figure 1: Plucking model decompositions for the U.S.
(a) Output, y t , and trend component, τ t
9.2
8.8
8.4
8.0
Output
Trend
7.6
55
60
65
70
75
80
85
90
95
00
95
00
(b) Transitory component, ct
0.01
0.00
-0.01
-0.02
-0.03
-0.04
55
60
65
70
75
80
85
90
(c) Probabilities of negative shocks to ct
1.0
0.8
0.6
0.4
0.2
0.0
55
60
65
70
75
80
85
90
95
00
16
Figure 2: Plucking model decompositions for the U.K.
(a) Output, y t , and trend component, τ t
12.0
11.8
11.6
11.4
11.2
Output
60
65
70
75
80
85
Trend
90
95
(b) Transitory component, ct
0.005
0.000
-0.005
-0.010
-0.015
-0.020
-0.025
60
65
70
75
80
85
90
95
(c) Probabilities of negative shocks to ct
1.0
0.8
0.6
0.4
0.2
0.0
60
65
70
75
80
85
90
95
17
Figure 3: Plucking model decompositions for Canada
(a) Output, y t , and trend component, τ t
2.4
2.2
2.0
1.8
1.6
1.4
1.2
1.0
Output
Trend
0.8
65
70
75
80
85
90
95
(b) Transitory component, ct
0.04
0.00
-0.04
-0.08
-0.12
-0.16
65
70
75
80
85
90
95
(c) Probabilities of negative shocks to ct
1.0
0.8
0.6
0.4
0.2
0.0
65
70
75
80
85
90
95
18
Figure 4: Plucking model decompositions for France
(a) Output, y t , and trend component, τ t
4.6
4.4
4.2
4.0
3.8
Output
Trend
3.6
70 72 74 76 78 80 82 84 86 88 90 92 94 96 98
(b) Transitory component, ct
0.00
-0.01
-0.02
-0.03
70 72 74 76 78 80 82 84 86 88 90 92 94 96 98
(c) Probabilities of negative shocks to ct
1.0
0.8
0.6
0.4
0.2
0.0
70 72 74 76 78 80 82 84 86 88 90 92 94 96 98
19
Figure 5: Plucking model decompositions for Germany
(a) Output, y t , and trend component, τ t
6.8
6.6
6.4
6.2
6.0
5.8
Output
Trend
5.6
65
70
75
80
85
90
95
(b) Transitory component, ct
0.00
-0.01
-0.02
-0.03
65
70
75
80
85
90
95
(c) Probabilities of negative shocks to ct
1.0
0.8
0.6
0.4
0.2
0.0
65
70
75
80
85
90
95
20
Figure 6: Plucking model decompositions for Italy
(a) Output, y t , and trend component, τ t
6.0
5.9
5.8
5.7
5.6
5.5
5.4
Output
Trend
5.3
76
78
80
82
84
86
88
90
92
94
96
98
96
98
(b) Transitory component, ct
0.01
0.00
-0.01
-0.02
-0.03
-0.04
-0.05
-0.06
76
78
80
82
84
86
88
90
92
94
(c) Probabilities of negative shocks to ct
1.0
0.8
0.6
0.4
0.2
0.0
76
78
80
82
84
86
88
90
92
94
96
98
21
Figure 7: Plucking model decompositions for Japan
(a) Output, y t , and trend component, τ t
13.5
13.0
12.5
12.0
11.5
Output
Trend
11.0
65
70
75
80
85
90
95
(b) Transitory component, ct
0.005
0.000
-0.005
-0.010
-0.015
65
70
75
80
85
90
95
(c) Probabilities of negative shocks to ct
1.0
0.8
0.6
0.4
0.2
0.0
65
70
75
80
85
90
95
22
23